Quantum isometries and group dual subgroups

Abstract

We study the discrete groups $Λ$ whose duals embed into a given compact quantum group, $\hatΛ\subset G$. In the matrix case $G\subset U_n^+$ the embedding condition is equivalent to having a quotient map $Γ_U\toΛ$, where $F=\{Γ_U|U\in U_n\}$ is a certain family of groups associated to $G$. We develop here a number of techniques for computing $F$, partly inspired from Bichon's classification of group dual subgroups $\hatΛ\subset S_n^+$. These results are motivated by Goswami's notion of quantum isometry group, because a compact connected Riemannian manifold cannot have non-abelian group dual isometries.